I think you need more info from the user, ideally with screenshots so you can see what is happening.

He says he is asked to log in (which suggests he's not ticking the stay logged in option) but then he says he needs to get a new password. Why is this? Is there a message telling him to do this? Or does he never remember his password and has to reset it each time?

When resetting a password, a user is sent one which is supposed to be temporary. I suspect this user is using this and then isn't changing his password to something more memorable once logged in and he isn't ticking the remain logged in option. That would explain why each time he visits, he has to log in and needs to reset his password as he can't remember it.